Frame Theoretical Derivation of Three Factor Learning Rule for Oja's Subspace Rule

arXiv stat.ML / 4/6/2026

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Key Points

The paper provides a principled derivation showing that an error-gated Hebbian learning rule for PCA (EGHR-PCA) is equivalent to Oja’s subspace rule under Gaussian inputs.
It uses frame theory to expand Oja’s subspace rule, explaining the origin of EGHR-PCA’s global third factor as a frame coefficient.
The work argues that the resulting three-factor rule is a non-heuristic, mathematically grounded route from a canonical learning rule to a more biologically plausible formulation.
Overall, it connects a modern three-factor learning formulation to established PCA dynamics through a formal geometric/mathematical framework.

Abstract

We show that the error-gated Hebbian rule for PCA (EGHR-PCA), a three-factor learning rule equivalent to Oja's subspace rule under Gaussian inputs, can be systematically derived from Oja's subspace rule using frame theory. The global third factor in EGHR-PCA arises exactly as a frame coefficient when the learning rule is expanded with respect to a natural frame on the space of symmetric matrices. This provides a principled, non-heuristic derivation of a biologically plausible learning rule from its mathematically canonical counterpart.